Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

COALESCENCE PHENOMENON OF QUANTUM COHOMOLOGY OF GRASSMANNIANS AND THE DISTRIBUTION OF PRIME NUMBERS

GIORDANO COTTI(†)

Abstract. The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grass- mannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly inﬂuenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypoth- esis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.

Contents Notations 2 1. Introduction and Results3 1.1. Plan of the paper. 6 2. Frobenius Manifolds and Quantum Cohomology6 2.1. Semisimple Frobenius Manifolds8 2.2. Gromov-Witten Theory and Quantum Cohomology9 3. Quantum Satake Principle 11 3.1. Results on classical cohomology of Grassmannians 11 3.2. Quantum Cohomology of G(k, n) 15 4. Frequency of Coalescence Phenomenon in QH•(G(k, n)) 16 4.1. Results on vanishing sums of roots of unity 17 4.2. Characterization of coalescing Grassmannians 18 4.3. Dirichlet series associated to non-coalescing Grassmannians, and their rareness 19 5. Distribution functions of non-coalescing Grassmannians, and equivalent form of the Riemann Hypothesis 22 References 24 arXiv:1608.06868v2 [math-ph] 21 Sep 2016

(†) SISSA, Via Bonomea, 265 - 34136 Trieste ITALY E-mail address: [email protected]. 1 • 2 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

Notations Given two natural numbers 0 < k < n we will denote by G(k, n) the Grassmannian of k-dimensional -subspaces of n. Thus (1, n) = n−1. C C G PC In what follows we will use the following notations for number theoretical functions: • P1(n) := min {p ∈ N: p is prime and p|n} , n ≥ 2; • for real positive x, y we deﬁne

Φ(x, y) := card ({n ≤ x: n ≥ 2,P1(n) > y}); P α • πα(n) := p prime p , α ≥ 0; p≤n • ζ(s) is the Riemann ζ-function; • ζ(s, k) will denote the truncated Euler product −1 Y 1 ζ(s, k) := 1 − , k ∈ , s ∈ \{0} . ps R>0 C p prime p≤k

• ζP (s) is the Riemann prime ζ-function, deﬁned on the half-plane Re(s) > 1 by the series X 1 ζ (s) := ; P ps p prime

• ζP,k(s) will denote the partial sums X 1 ζ (s) := ; P,k ps p prime p≤k

• ω : R≥1 → R is the Buchstab function ([Buc37]), i.e. the unique continuous solution of the delay diﬀerential equation d (uω(u)) = ω(u − 1), u ≥ 2, du with the initial condition 1 ω(u) = , for 1 ≤ u ≤ 2. u

If f, g : R+ → R, with g deﬁnitely strictly positive, we will write • f(x) = Ω+(g(x)) to denote f(x) lim sup > 0; x→∞ g(x) • f(x) = Ω−(g(x)) to denote f(x) lim inf < 0; x→∞ g(x)

• f(x) = Ω±(g(x)) if both f(x) = Ω+(g(x)) and f(x) = Ω−(g(x)) hold. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 3

1. Introduction and Results In this paper we exhibit a direct connection between the theory of Frobenius Manifolds, more precisely the Gromov-Witten and Quantum Cohomology theories, and one of the most ancient problems lying at the heart of Mathematics, the distribution of prime numbers. We study a phenomenon of resonance in the small quantum cohomology of complex Grassmannians (consisting in the coalescence of some natural parameters there deﬁned), and show that the occurrence and frequency of this phenomenon is surprisingly related to the distribution of prime numbers. This relation is so strict that it leads to (at least) two equivalent formulations of the famous Riemann Hypothesis: the former is given as a constraint on the disposition of the singularities of a generating function of the numbers of Grassmannians not presenting the resonance, the latter as an (essentially optimal) asymptotic estimate for a distribution function of the same kind of Grassmannians. Besides their geometrical-enumerative meaning, three point genus zero Gromov-Witten invariants of complex Grassmannians implicitly contain information about the distribution of prime numbers. This mysterious relation deserves further investigations.

Born in the last decades of the XX-th century, in the middle of the creative impetus for a mathemat- ically rigorous foundations of Mirror Symmetry, the theory of Frobenius Manifolds ([Dub96], [Dub98], [Dub99], [Man99], [Her02], [Sab08]) seems to be characterized by a sort of universality (see [Dub04]): this theory, in some sense, is able to unify in a unique, rich, geometrical and analytical description many aspects and features shared by the theory of Integrable Systems, Singularity Theory, Gromov-Witten In- variants, the theory of Isomondromic Deformations and Riemann-Hilbert Problems, as well as the theory of special functions like Painlevé Transcendents.

Originally introduced by physicists ([Vaf91]), in the context of N = 2 Supersymmetric Field Theo- ries and mirror phenomena, the Quantum Cohomology of a complex projective variety X (or more in general a symplectic manifold [MS12]) is a family of deformations of its classical cohomological algebra • L k • structure defined on H (X) := k H (X; C), and parametrized over an open domain D ⊆ H (X): the ∼ • fiber over p ∈ D is identified with the tangent space TpD = H (X). This is exactly the prototype of a Frobenius manifolds, namely a manifold, endowed with a flat metric, on whose tangent spaces there is a well defined commutative, associative, unitary algebra structure, satisfying also some more compatibility conditions. The structure constants of the quantum deformed algebras are given by (third derivatives of) X a generating function F0 of Gromov-Witten Invariants of genus 0 of X: these rational numbers morally “count” (modulo parametrizations) algebraic/pseudo-holomorphic curves of genus 0 on X, with a fixed degree, and intersecting some fixed subvarieties of X.

Focusing on the case of complex Grassmannians, X = G(k, n), in what follows we study the occurrence of a phenomenon of coalescence of some numerical parameters on their small quantum cohomology (i.e. D ∩ H2(X)). This set of parameters defines a system of coordinates (called Dubrovin canonical coordinates) near any point whose corresponding Frobenius algebra is semisimple. It is important to keep in mind that these parameters have a double algebro-geometric meaning: (1) at each point p ∈ D their vector fields coincide with the idempotents of the algebra structure defined on TpD; (2) they are the eigenvalues1 of an operator of multiplication by a distinguished vector field (the Euler vector field) which codifies the grading structure of the algebras. Notice that, along the small quantum locus, the Euler vector field coincides with the first Chern class c1(X). Moreover, this coalescence of canonical coordinates can be interpreted as a “delicate point”2 in the local description of semisimple Frobenius manifolds as spaces of deformation parameters for isomonodromic families of linear differential systems with rational coefficients in complex domains, at least as exposed

1The canonical coordinates are not uniquely determined by the requirement (1) alone: there is a shift ambiguity. We ﬁx this freedom by requirement (2). 2In the theory of Painlevé equations they are called critical points. • 4 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS in [Dub98] and [Dub99]. Such a description is carried on in all details in [CG16], for the general analytic case, and in [CDG16] for the speciﬁc Frobenius case. For simplicity, we will call coalescing a Grassman- nian such that some of the Dubrovin canonical coordinates coalesce.

The questions, to which we answer in the present paper, are the following: (i) For which k, n the Grassmannian G(k, n) is coalescing? (ii) How frequent is the phenomenon of coalescence among all Grassmannians?

Let us summarize some of the main results obtained.

Theorem (cf. Theorems 4.4, 4.5, 5.1, 5.2 and Corollaries 4.3, 4.2 for more details)

PART I The complex Grassmannian G(k, n) is coalescing if and only if P1(n) ≤ k ≤ n − P1(n). In particular, all Grassmannians of proper subspaces of Cp, with p prime, are not coalescing.

PART IILet us denote by lñ, for n ≥ 2, the number of non-coalescing Grassmannians of proper subspaces of Cn, i.e. lñ := card {k : G(k, n) is not coalescing} , and let ∞ X lñ Le (s) := ns n=2 be the associated Dirichlet series. Le (s) is absolutely convergent in the half-plane Re(s) > 2, where it can be represented by the infinite series involving the Riemann zeta function and the truncated Euler products X p − 1 2ζ(s) Le (s) = − 1 . ps ζ(s, p − 1) p prime

By analytic continuation, Le (s) can be extended to (the universal cover of) the punctured half-plane ( ) ρ ρ pole or zero of ζ(s), {s ∈ : Re(s) > σ}\ s = + 1: , C k k squarefree positive integer

  1  X  3 σ := lim sup · log  l˜k , 1 ≤ σ ≤ , n→∞ log n 2  k≤n  k composite having logarithmic singularities at the punctures.

In particular, we have the equivalence of the following statements: 1 • (RH) all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(s) = 2 ; 0 • the derivative Le (s) extends, by analytic continuation, to a meromorphic function in the half- 3 plane 2 < Re(s) with a single pole of oder one at s = 2.

At the point s = 2 the following asymptotic estimate holds 1 Le (s) = log + O(1), s → 2, Re(s) > 2. s − 2 • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 5

As a consequence, we have that n X 1 n2 l˜ ∼ , k 2 log n k=2 which means that non-coalescing Grassmannians are rare.

Figure 1. In this ﬁgure we represent complex Grassmannians as disposed in a Tartaglia- Pascal triangle: the k-th element (from the left) in the n-th row (from the top of the triangle) represents the Grassmannian G(k, n + 1), where n ≤ 102. The dots colored in black represent non-coalescing Grassmannians, while the dots colored in gray the coalescing ones. The reader can note that black dots are rare w.r.t. the gray ones, and that the black lines correspond to Grassmannians of subspaces in Cp with p prime.

PART III For x ∈ R≥4 we deﬁne ( ) 2 ≤ n ≤ x is such that G(k, n) Hb (x) := card n: 1 . is not coalescing for 1 ≤ k ≤ [x 2 ] + 1 We have the following results: (a) for any κ > 1, the following integral representations3 hold    1 Z ζ(s) xs Hb (x) =  − 1 − ζ 1 (s) + ζ 1 (s) ds, 2πi 1 P,2x 2 +1 P,x 2 +1 s Λκ ζ s, x 2 + 1    Z 1 ζ(s) s 1 s 1 s ds Hb (x) =  − 1 x + ζP (s) (x 2 + 1) − (2x 2 + 1)  , 2πi 1 s Λκ ζ s, x 2 + 1

both valid for x ∈ R≥2 \ N, and where Λκ := {κ + it: t ∈ R} is the line oriented from t = −∞ to t = +∞.

3The integral must be interpreted as a Cauchy Principal Value. • 6 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

(b) The function Hb admits the following asymptotic estimate: Z x dt Θ Hb (x) = + O x log x , where Θ := sup {Re(ρ): ζ(ρ) = 0} . 0 log t 1 Hence, it is clear the equivalence of (RH) with the (essentially optimal) estimate with Θ = 2 .

Question (i) has already been addressed in [GGI16] (Remark 6.2.9): it is claimed, but not proved, that the condition gcd(min(k, n − k)!, n) > 1 (which is equivalent to the condition P1(n) ≤ k ≤ n − P1(n)) is a necessary condition for coalescence of some canonical coordinates in the small quantum locus of G(k, n). 1.1. Plan of the paper. In Section 2 we summarize some basic notions about Frobenius Manifolds, Gromov-Witten Theory and Quantum Cohomology. In Section 3 we expose a description, valid both in the classical and in the quantum setup, of the cohomology of Grassmannians as alternate products of cohomology of Projective Spaces. In Section 4 the coalescence phenomenon is completely characterized, and a generating function (Dirichlet series) for the numbers of non-coalescing Grassmannians is introduced: from the study of the behavior of this function near its singularities, we deduce the rareness (i.e. zero density) of the non-coalescence. In Section 5 we introduce and study some distribution functions for non-coalescing Grassmannians: some integral representations and asymptotic expansions are found. It is shown that RH can be reformulated as an (essentially optimal) estimate for one of the distribution function introduced.

Acknowledgements. The author is grateful to Boris Dubrovin and Davide Guzzetti for useful discus- sions and encouragement, to Alberto Perelli and Gérald Tenenbaum for the friendly e-mail conversations, to Don Zagier for critical comments, and to Michael Zieve for pointing out a Theorem due to H.B. Mann (Theorem 4.1), used in the proof of Proposition 4.1.

2. Frobenius Manifolds and Quantum Cohomology Introduced and extensively developed by B. Dubrovin in [Dub98], [Dub96], [Dub99] in order to give a differential geometrical description of the WDVV-system of equations obtained in two dimensional topological field theories ([DVV91], [Wit90]), Frobenius Manifolds are, roughly speaking, smooth/analytic manifolds whose tangent spaces are endowed with an associative, commutative and unitary algebra4 structure, smoothly/analytically depending on the point. Furthermore, in order to be Frobenius, these algebras structures must satisfy a compatibility condition with respect to a symmetric bilinear non- degenerate form, simply called metric5: if M is a Frobenius manifold, with metric η, and we denote by ◦p the product defined on the tangent space TpM at the point p ∈ M, then the required compatibility condition is η(a ◦p b, c) = η(a, b ◦p c) for all a, b, c ∈ TpM, p ∈ M. (2.1) In the remaining part of the paper, we will consider only complex analytic Frobenius manifolds. It could be useful for the reader to keep in mind the following elementary example of Frobenius algebra. Example 2.1. Let X be a smooth compact manifold of even dimension, with vanishing odd cohomology, 2k+1 ∼ L k i.e. H (X) = 0 for all k ≥ 0. Let us consider its classical cohomology ring A := H (X; C), endowed with the ∪-product, and the Poincaré metric η defined by Z η(α, β) := α ∪ β. X Then, the resulting algebra structure is associative, commutative, unitary, there is a well defined C- bilinear symmetric form which is non-degenerate (Poincaré Duality Theorem), with respect to which

4 In the smooth and real analytic cases, the ground field of the algebra is R, in the complex analytic case (on which we will focus in the present paper) is C. 5If M is a complex analytic Frobenius manifold, the metric is defined on the holomorphic tangent bundle TM, and it is just OM -bilinear. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 7 the compatibility condition (2.1) is trivially satisfied. A is the classical Frobenius cohomological algebra associated to the manifold X. From a physical point of view, this algebra describes the matter sector of the topological σ-model with target space X. One of the main features of the algebra A is that it is a naturally Frobenius graded algebra: there exists a linear morphism Q: A → A, called grading operator, such that Q(α ∪ β) = Q(α) ∪ β + α ∪ Q(β), η(Q(α), β) + η(α, Q(β)) = d · η(α, β), where d is a number called the charge of the algebra. In Example 2.1, the operator Q is defined as the 1 1 map acting on a homogeneous basis (Ti)i of A as Q(Ti) = 2 deg(Ti) · Ti, and the charge is d = 2 dimR X. If we restrict to the case of Frobenius algebras with diagonalizable grading operators, by imposing a flatness condition for the Levi-Civita connection associated to η, we can extend the grading structure on a whole Frobenius manifold by the choice of a distinguished affine vector field, the so-called Euler vector field. We give now the complete and detailed definition. Definition 2.1. A Frobenius manifold structure on a complex manifold M of dimension n is defined by giving (FM1) a symmetric O(M)-bilinear metric tensor η ∈ Γ J2 T ∗M , whose corresponding Levi-Civita connection ∇ is flat; (FM2)a (1, 2)-tensor c ∈ Γ TM ⊗ J2 T ∗M such that • c[ ∈ Γ J3 T ∗M , •∇ c[ ∈ Γ J4 T ∗M ; (FM3) a vector field e ∈ Γ(TM), called the unity vector field, such that • the bundle morphism c(−, e, −): TM → TM is the identity morphism, •∇ e = 0; (FM4) a vector field E ∈ Γ(TM), called the Euler vector field, such that • LEc = c, • LEη = (2 − d) · η, where d ∈ C is called the charge of the Frobenius manifold. The tensor c is the tensor of constants structure of a Frobenius algebra on each tangent space: the multiplication of vector fields will be denoted by ◦. We leave as an easy exercise for the reader to deduce from the axioms above that the resulting algebra on each tangent space is Frobenius. Being the connection α ∇ flat, there exist local flat coordinates, that we denote (t )α, w.r.t. which the metric η is constant. Moreover, observe that because of flatness and the conformal Killing condition, the Euler vector field is affine, i.e. ∇∇E = 0. Without loss of generality, we can mark the first flat coordinate t1 in such a way that ∂ • ∂t1 coincides with the unity vector field, P ν ν ν • and that the local expression of the Euler vector field is E = ν [(1 − qν )t + r ] ∂ν , with qν , r ∈ ν C, q1 = 0 and r 6= 0 if and only if qν = 1. α Since the metric η is flat, in flat local coordinates (t )α the connection ∇ coincides with partial derivatives: [ J4 ∗ so, the condition ∇c ∈ Γ T M means that ∂αcβγδ is symmetric in all indices. This implies the local existence of a function F such that

cαβγ = ∂α∂β∂γ F. The associativity of the algebra, together with its graded structure, are equivalent to the following conditions for F , called WDVV-equations: γδ γδ Fαβγ η Fδν = Fνβγ η Fδα,Fαβγ := ∂α∂β∂γ F. (2.2)

ηαβ = F1αβ, LEF = (3 − d) · F + Q(t), (2.3) • 8 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS where Q(t) denotes a quadratic polynomial in tα’s. Giving a solution of the WDVV problem (2.2), (2.3) is equivalent to locally defining a Frobenius manifold structure (for more details see [Dub96], [Dub99]). 2.1. Semisimple Frobenius Manifolds. We will focus now on a particularly interesting and vast class of Frobenius manifolds, the class of semisimple Frobenius manifolds. Definition 2.2. A point p of a Frobenius manifold M, of complex dimension n, is said to be a semisimple point if the corresponding Frobenius algebra (TpM, η|p, ◦p) satisfies one of the following equivalent conditions: (1) it is without nilpotents; (2) it is isomorphic to Cn; (3) it admits a basis of orthogonal idempotent vectors ϕ1, . . . , ϕn:

ϕi ◦p ϕj = δijϕi, ηp(ϕi, ϕj) = ηp(ϕi, ϕi)δij;

(4) it admits a regular vector, i.e. a vector v such that the operator v ◦p (−): TpM → TpM has simple spectrum. A Frobenius manifold with an open dense subset of semisimple points is called semisimple6. Locally, near semisimple points, a distinguished system of (non-ﬂat) coordinates is always well-deﬁned:

Theorem 2.1 (B. Dubrovin [Dub99]; see also [CDG16]). Let M be a Frobenius manifold, and p0 ∈ M a semisimple point. On any sufficiently small, simply-connected, open neighborhood U of p0, made of semisimple points, a coherent labeling (u1, . . . , un) of the eigenvalues of the operators U := E ◦p (−): TpM → TpM, p ∈ U, can be used as holomorphic local coordinates such that the coordinate vector fields ∂ ∂ ,..., ∂u1 ∂un are the orthogonal idempotent vector fields at any point p ∈ U. We will refer to this particular system of coordinates as the system of Dubrovin canonical coordinates. One of the main features of the theory of semisimple Frobenius manifolds is that their whole structure can be locally parametrized by a finite number of parameters, or local moduli. A particularly effective method of local parametrization of semisimple Frobenius structures consists in their (local) identification with spaces of independent deformation parameters for isomonodromic families of equations in complex domains with rational coefficients. If M is a Frobenius manifold, and p0 ∈ M is a point with pairwise distinct canonical coordinates (i.e. ui(p0) 6= uj(p0) for i 6= j; this implies that p0 is semisimple), then, for a sufficiently small neighborhood N of p0, we can associate to any point p ∈ N a differential system on P1(C) \{0, ∞} of the form dY V (p) = U(p) + Y,Y (z) ∈ M ( ), z ∈ [\ 0, (2.4) dz z n C C such that

• U(p) := diag(u1(p), . . . , un(p)), • the matrix V (p) is anti-symmetric, • the monodromy data of solutions of the equation do not depend on p. We refer the reader to [Dub96],[Dub99] and [Guz01] for more details, and explicit formulae for the local reconstruction of the Frobenius structure starting from the monodromy data of the isomonodromic family (2.4). The extension of this local description of Frobenius manifolds, as spaces of deformation parameters of isomonodromic families of equations, near semisimple points with coalescing canonical coordinates, is

6A suﬃcient condition for a complex analytic Frobenius manifold to be semisimple is to have at lest one semisimple point: it is clear, indeed, that the semisimplicity condition is open. For a proof of the density of the open set of semisimple points see [Her02]. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 9 quite delicate, and it is treated in all details in [CG16] (for the general analytic theory) and in [CDG16] (for the speciﬁc case Frobenius).

2.2. Gromov-Witten Theory and Quantum Cohomology. Let X be a smooth projective complex variety. In order not to deal with superstructures, we will suppose for simplicity that the variety X 2k+1 ∼ has vanishing odd cohomology, i.e. H (X; C) = 0 for 0 ≤ k. Let us ﬁx a homogeneous basis • L 2k (T0,T1,...,TN ) of H (X; C) = k H (X; C) such that • T0 = 1 is the unity of the cohomology ring; 2 • T1,...,Tr span H (X; C). We will denote by η : H•(X; C) × H•(X; C) → C the Poincaré metric Z η(ξ, ζ) := ξ ∪ ζ, X and in particular Z ηαβ := Tα ∪ Tβ. X If β ∈ H2(X; Z)/torsion, we denote by Mg,n(X, β) the Kontsevich-Manin moduli stack of n-pointed, genus g stable maps to X of degree β, which parametrizes:

• equivalence classes of pairs ((Cg, x); f), where (Cg, x) is an n-pointed algebraic curve of genus g, with at most nodal singularities and with n marked points x = (x1, . . . , xn), and f : Cg → X 0 0 0 is a morphism such that f∗[Cg] ≡ β. Two pairs ((Cg, x); f) and ((Cg, x ); f ) are deﬁned to be 0 0 equivalent if there exists a bianalytic map ϕ: Cg → Cg such that ϕ(xi) = xi, for all i = 1, . . . , n, and f 0 = ϕ ◦ f. • The morphisms f are required to be stable: if f is constant on any irreducible component of Cg, then that component should have only a ﬁnite number of automorphisms as pointed curves (in other words, it must have at least 3 distinguished points, i.e. points that are either nodes or marked ones).

We will denote by evi : Mg,n(X, β) → X : ((Cg, x); f) 7→ f(xi) the naturally defined evaluations maps, 2 and by ψi ∈ H (Mg,n(X, β); Q) the Chern classes of tautological cotangent line bundles L → M (X, β), L | = T ∗ C , ψ := c (L ). i g,n i ((Cg ,x);f) xi g i 1 i virt Using the construction of [BF97] of a virtual fundamental class [Mg,n(X, β)] in the Chow ring A∗(Mg,n(X, β)), and of degree equal to the expected dimension Z virt [Mg,n(X, β)] ∈ AD(Mg,n(X, β)),D = (1 − g)(dimC X − 3) + n + c1(X), X a good theory of intersection is allowed on the Kontsevich-Manin moduli stack. We can thus define the Gromov-Witten invariants (with descendants) of genus g, with n marked points and of degree β of X as the integrals (whose values are rational numbers) Z n X Y ∗ di hτd1 γ1, . . . , τdn γnig,n,β := evi (γi) ∪ ψi , (2.5) virt [Mg,n(X,β)] i=1 • γi ∈ H (X; C), di ∈ N, i = 1, . . . , n. Since by effectiveness (see [Man99], [KM94]) the integral is non-vanishing only for effective classes β ∈ Eff(X) ⊆ H2(X; Z), the generating function of rational numbers (2.5), called total descendent potential (or also gravitational Gromov-Witten potential, or even Free Energy) of genus g is defined as the formal series ∞ X X Qβ F X (γ, Q) := hγ. . . . , γiX , g n! g,n,β n=0 | {z } β∈Eff(X) n times • 10 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

α,p where we have introduced (inﬁnitely many) coordinates t := (t )α,p

X α,p γ = t τpTα, α = 0, . . . , N, p ∈ N, α,p and formal parameters R R T1 Tr β β β Q := Q1 ····· Qr ,Qi’s elements of the Novikov ring Λ := C[[Q1,...,Qr]]. X The free energy Fg ∈ Λ[[t]] can be seen a function on the large phase-space, and restricting the free energy to the small phase space (naturally identiﬁed with H•(X; C)), X 1,0 N,0 X Fg (t , . . . , t ) := Fg (t)|tα,p=0, p>0, one obtains the generating function of the Gromov-Witten invariants of genus g. X By the Divisor axiom, the genus 0 Gromov-Witten potential F0 (t), can be seen as an element of the 0 t1 tr r+1 N X ring C[[t ,Q1e ,...,Qre , t , . . . , t ]]: in what follows we will be interested in cases in which F0 is the analytic expansion of an analytic function, i.e.

X n 0 t1 tr r+1 N o F0 ∈ C t ,Q1e ,...,Qre , t , . . . , t .

X Without loss of generality, we can put Q1 = Q2 = ··· = Qr = 1, and F0 (t) deﬁnes an analytic function in an open neighborhood D ⊆ H•(X; C) of the point ti = 0, i = 0, r + 1,...,N, (2.6) Re ti → −∞, i = 1, 2, . . . , r. (2.7)

X The function F0 is a solution of WDVV equations (for a proof see [KM94], [Man99], [CK99]), and thus it defines an analytic Frobenius manifold structure on D. Definition 2.3. The Frobenius manifold structure defined on the domain of convergence D of the X Gromov-Witten potential F0 , solution of the WDVV problem, is called Quantum Cohomology of X, and denoted by QH•(X). Note that • the flat metric is given by the Poincaré metric η; • the unity vector field is T0 = 1, using the canonical identifications of tangent spaces ∼ • TpD = H (X; C): ∂tα 7→ Tα; • the Euler vector field is N X 1 E := c (X) + 1 − deg T tαT . 1 2 α α α=0 By the expression small quantum cohomology (or small quantum locus) we denote the Frobenius structure attached to points in D ∩ H2(X; C). Remark 2.1. We have no general results guaranteeing the converge of the Gromov-Witten potential for a generic smooth projective variety X; however, for some classes of varieties, it is known that the sum X 0 r+1 N defining F0 at points with coordinates t = t = ··· = t = 0 is finite. This is the case for • Fano varieties, • varieties admitting a transitive action of a semisimple Lie group. For a proof see [CK99]. Notice that for these varieties the small quantum locus coincide with the whole 2 X space H (X; C). Conjecturally, for Calabi-Yau manifolds the series defining F0 is convergent in a neighborhood of the classical limit point (see [CK99], [KM94]). In case of convergence, the potential (and hence the whole Frobenius structure) can be maximally analytically continued to an unramified covering of a subdomain of H•(X; C). We refer to this global Frobenius structure as the Big Quantum Cohomology. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 11

Remark 2.2. At the classical limit point (2.6), (2.7), the algebra structure on the tangent spaces coincide with the classical cohomological algebra structure introduced in Example 2.1. Indeed, the following is the structure of the potential: (1) by Point Mapping Axiom, the Gromov-Witten potential can be decomposed into a classical term and a quantum correction as follows

X F0 (γ) = Fclassical + Fquantum ∞ N 1 Z X X 1 X = γ3 + hγ, . . . , γiX , where γ = tαT ; 6 k! 0,k,β α X k=0 | {z } α=0 β∈Eﬀ(X)\{0} k times

0 X (2) the variable t appears only in the classical term of F0 ; (3) because of the Divisor axiom, the variables corresponding to cohomology degree 2 (i.e. t1, . . . , tr) appear in the exponential form in the quantum term; the Frobenius structure is 2πi-periodic in the 2-nd cohomology directions: the structure can be considered as deﬁned on an open region of the quotient H•(X; C)/2πiH2(X; Z). Remark 2.3. The Frobenius algebra of Example 2.1 is not semisimple (it clearly contains nilpotents). By quantum deformation of the ∪-product, it may happen that the quantum cohomology is semisimple. There are no general results characterizing smooth projective varieties with semisimple quantum cohomology: however, for some classes of varieties such as • some Fano threefolds [Cio04], • toric varieties [Iri07], • some homogeneous spaces [CMP10], it has been proved that the small quantum locus is made of semisimple points. Grassmannians are among these varieties. More general homogeneous spaces may have non-semisimple small quantum cohomology ([CMP10], [CP11]). Some necessary conditions for semisimplicity are given in [HMT09]; some suﬃcient conditions for other Fano varieties are given in [Per]. There is an intriguing conjecture ([Dub98], [Dub13]) which relates the enumerative geometry of a projective variety X with semisimple quantum cohomology with its derived category of coherent sheaves Db(X): some more details and new results can be found in [CDG16] and [CDG].

3. Quantum Satake Principle The quantum cohomology of Grassmannians has been one of the first cases that both physicists [Wit95] and mathematicians (see e.g. [Ber96], [Ber97], [Buc03]) studied in details. In this section we expose an identification, valid both in the classical ([Mar00]) and in the quantum setup ([GM], [BCFK05], [GGI16]), of the cohomology of Grassmannians with an alternate product of the cohomology of Projective Spaces. This identification has been well known to physicists for long time: e.g. the reader can find an analogue description of the supersymmetric σ-model of G(k, n) in Section 8.3 and Appendix A of the paper [CV93], on the classification of N = 2 Supersymmetric Field Theories. In the context of the theory of Frobenius manifolds, such an identification has been generalized and axiomatized in [KS08] in the notion of alternate product of Frobenius manifolds.

3.1. Results on classical cohomology of Grassmannians. A classical reference for cohomology of Grassmannians is [GH78]. Let us introduce the following notations, used only in this section, to denote the (products of) complex ﬂag manifolds

n−1 P := P , Π := P × · · · × P C | {z } k times

G := G(k, n), F := Fl(1, 2, . . . , k, n), • 12 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS with n ≥ 2, and 1 ≤ k < n. If we ﬁx an hermitian structure on Cn (e.g. the standard one), we have the following diagram: F o p ι ~ G Π where p is the canonical projection, and ι the inclusion. Moreover, it is also deﬁned a natural rational map «taking the span»

Π / G:(`1, . . . , `k) 7→ spanh`1, . . . , `ki , whose domain is the image of ι. On the manifold Π we have k canonical line bundles, denoted Lj for j = 1, . . . , k, deﬁned as the pull-back of the bundle O(1) on the j-th factor P. If we denote V1 ⊆ V2 ⊆ · · · ⊆ Vk the tautological bundles over F, we have that ∗ ∼ ∨ ι Lj = (Vj/Vj−1) . ∗ ∗ Denoting with the same symbol xi the Chern class c1(Li) on Π and its pull-back c1(ι Li) = ι c1(Li) on F, we have • ∼ • ⊗k ∼ C[x1, . . . , xk] H (Π; C) = H (P; C) = n n (by Künneth Theorem), hx1 , . . . xk i • ∼ C[x1, . . . , xk] H (F; C) = , hhn−k+1, . . . , hni where hj stands for the j-th complete symmetric polynomial in x1, . . . , xk. Since the classes x1, . . . , xk are the Chern roots of the dual of the tautological bundle Vk, we also have

Sk • ∼ C[e1, . . . , ek] ∼ C[x1, . . . , xk] H (G; C) = = , hhn−k+1, . . . , hni hhn−k+1, . . . , hni where the ej’s are the elementary symmetric polynomials in x1, . . . , xk. This is the classical representation of the cohomology ring of the Grassmannian G with generators the Chern classes of the dual of the tautological vector bundle S, and relations generated by the Segre classes of S. From this ring representation, it is clear that any cohomology class of G can be lifted to a cohomology class of Π: we will say that γ˜ ∈ H•(Π; C) is the lift of γ ∈ H•(G; C) if p∗γ = ι∗γ˜. The following integration formula allow us to express the cohomology pairings on H•(G; C) in terms of the cohomology pairings on H•(Π; C). Theorem 3.1 (S. Martin, [Mar00]). If γ ∈ H•(G; C) admits the lift γ˜ ∈ H•(Π; C), then k Z (−1)(2) Z γ = γ˜ ∪ ∆2, (3.1) k! Π G Π where Y ∆ := (xi − xj). 1≤i

H•( ; )⊗k π / Vk H•( ; ) P O C ? P C i v ? j [H•(P; C)⊗k]ant where k • ⊗k ^ • π : H (P; C) → H (P; C): α1 ⊗ · · · ⊗ αk 7→ α1 ∧ · · · ∧ αk,

k ^ • • ⊗k ant X i: H (P; C) → [H (P; C) ] : α1 ∧ · · · ∧ αk 7→ ε(ρ)αρ(1) ⊗ · · · ⊗ αρ(k), ρ∈Sk together with its inverse ^k 1 j :[H•( ; )⊗k]ant → H•( ; ): α ⊗ · · · ⊗ α 7→ α ∧ · · · ∧ α . P C P C 1 k k! 1 k The Poincaré pairing gP on H•(P; C) induces a metric g⊗P on H•(P; C)⊗k and a metric g∧P on Vk • H (P; C) given by k ⊗ Y g P(α1 ⊗ · · · ⊗ αk, β1 ⊗ · · · ⊗ βk) := gP(αi, βi), i=1 ∧P P g (α1 ∧ · · · ∧ αk, β1 ∧ · · · ∧ βk) := det g (αi, βj) 1≤i,j≤k . Using the identiﬁcations above, when g⊗P is restricted on the subspace [H•(P; C)⊗k]ant it coincides with ∧ Vk • k!g P on H (P; C). We deduce the Corollary 3.2. The isomorphism k • ^ • (k) ∧ j ◦ ϑ: H (G; C), gG → H (P; C), (−1) 2 g P is an isometry.

Proof. It follows immediately from the integration formula (3.1). An additive basis of H•(G; C) is given by the Schubert classes (Poincaré-dual to the Schubert cycles), given in terms of x1, . . . , xk by the Schur polynomials

 λ1+k−1 λ2+k−2 λk  x1 x1 . . . x1 λ1+k−1 λ2+k−2 λk x2 x2 . . . x2  det    .   .  xλ1+k−1 xλ2+k−2 . . . xλk σ := k k k λ  k−1 k−2  x1 x1 ... 1 k−1 k−2 x2 x2 ... 1 det    .   .  k−1 k−2 xk xk ... 1 where λ is a partition whose corresponding Young diagram is contained in a k × (n − k) rectangle. The • lift of each Schubert class to H (Π; C) is the Schur polynomial in x1, . . . , xk (indeed each xi in the Schur • 14 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS polynomial has exponent at most n − k < n). Thus, under the identiﬁcation above, the class j ◦ ϑ(σλ) is Vk • 2 σλ1+k−1 ∧ · · · ∧ σλk ∈ H (P; C), σ being the generator of H (P; C). • ∼ • ⊗k Using the Künneth isomorphism H (Π; C) = H (P; C) , the cup product ∪Π is expressed in terms of ∪P as follows: !   X i i X j j X i j i j α1 ⊗ · · · ⊗ αk ∪Π  β1 ⊗ · · · ⊗ βk = (α1 ∪P β1) ⊗ · · · ⊗ (αk ∪P βk). i j i,j

• S • • If γ ∈ H (Π; C) k , then γ ∪Π (−): H (Π; C) → H (Π; C) leaves invariant the subspace of anti- Vk • symmetric classes. Thus, γ ∪Π (−) induces an endomorphism Aγ ∈ End H (P; C) that acts on decomposable elements α = α1 ∧ · · · ∧ αk as follows 1 X A (α) = j(γ ∪ i(α)) = ε(ρ)(γi ∪ α ) ∧ · · · ∧ (γi ∪ α ), (3.2) γ Π k! 1 P ρ(1) k P ρ(k) i,ρ i • where γj ∈ H (P; C) are such that X i i γ = γ1 ⊗ · · · ⊗ γk. i Vk • As an example, in the following Proposition we reformulate in H (P; C) the classical Pieri formula, • expressing the multiplication by a special Schubert class σ` in H (G; C) X σ` ∪G σµ = σν , ν where the sum is on all partitions ν which belong to the set µ⊗` (the set of partitions obtained by adding ` boxes to µ, at most one per column) and which are contained in the rectangle k × (n − k), in terms of ` • the multiplication by σ` = (σ) ∈ H (P; C). We also make explicit the operation of multiplication by the • classes p` ∈ H (G; C) deﬁned in terms of the special Schubert classes by   `(m + ··· + m − 1)! k  X 1 k Y mi  p` := −  (−σi)  , ` = 0, . . . , n − 1,  m1! . . . mk!  m1+2m2+···+kmk=` i=1 m1,...,mk≥0 • because of the nice form of their lifts p˜` ∈ H (Π; C). • Proposition 3.1. If σµ ∈ H (G; C) is a Schubert class, then

• the product σ` ∪G σµ with a special Schubert class σ` is given by   k 1  X X ^  j ◦ ϑ(σ` ∪ σµ) =  σi ∪ σµj +k−j ; G k!  ρ(j) P  i1+···+ik=` ρ∈Sk j=1 i1,...,ik≥0

• the product p` ∪G σµ is given by k X j ◦ ϑ(p` ∪G σµ) = σµ1+k−1 ∧ · · · ∧ (σµi+k−i ∪P σ`) ∧ · · · ∧ σµk . i=1 Proof. From Corollary (3.1) we have

ϑ(σ` ∪G σµ) =σ ˜` ∪Π ϑ(σµ) • If γ =σ ˜` is the lift of the special Schubert class σ` ∈ H (G; C), then X σ˜` = h`(x1, . . . , xk) = σi1 ⊗ · · · ⊗ σik ,

i1+···+ik=` i1,...,ik≥0 • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 15 and using (3.2) we easily conclude. Analogously, we have that

k k X ` X p˜` = xi = 1 ⊗ · · · ⊗ σ` ⊗ · · · ⊗ 1, i-th i=1 i=1 and k X Ap˜` (α) = α1 ∧ · · · ∧ (σ` ∪P αi) ∧ · · · ∧ αk. i=1 3.2. Quantum Cohomology of G(k, n). The identiﬁcation in the classical cohomology setting of • Vk • H (G; C) with the wedge product H (P; C), exposed in the previous section, has been extended also to the quantum case in [BCFK05], [BCFK08], [CFKS08], and [KS08]. The following isomorphism of the (small) quantum cohomology algebra of Grassmannians at a point 2 tσ1 = log q ∈ H (G; C) is well-known

Sk • ∼ C[x1, . . . , xk] [q] QHq (G) = k−1 , hhn−k+1, . . . , hn − (−1) qi while for the (small) quantum cohomology algebra of Π, being equal to the k-fold tensor product of the quantum cohomology algebra of P, we have [x , . . . , x ][q , . . . , q ] QH• (Π) ∼ C 1 k 1 k . q1,...,qk = n n hx1 − q1, . . . , xk − qki Following [BCFK05], and interpreting now the parameters q’s just as formal parameters, if we denote • by QH (Π) the quotient of QH• (Π) obtained by substituting q = (−1)k−1q, and denoting the q q1,...,qk i canonical projection by • [−] : QH• (Π) → QH (Π), q q1,...,qk q we can extend by linearity the morphisms ϑ, j of the previous section to morphisms • • ϑ: QHq (G) → QHq (Π), ant k h • i ^ • j : QHq (Π) → H (P; C) ⊗C C[q].

Notice that the image under ϑ of any Schubert class σλ is equal to the classical product σ˜λ ∪Π ∆, the Q exponents of xi’s in the product σλ(x) i

• Theorem 3.2 ([BCFK05]). For any Schubert classes σλ, σµ ∈ H (G; C) we have

ϑ(σλ ∗G,q σµ) = [ϑ(σµ) ∗Π,q1,...,qk σ˜λ]q . (3.3) In particular, using the identiﬁcation j, we have that k X j ◦ ϑ(σ ∗ p ) = σ ∧ · · · ∧ σ ∗ k−1 σ ∧ · · · ∧ σ . (3.4) µ G,q ` µ1+k−1 µi+k−i P,(−1) q ` µk i=1 Proof. The essence of the result (3.3) is the following identity between 3-point Gromov-Witten invariants of genus 0 of the grassmannian G and Π: k (2) (−1) X d(k−1) Π G ∨ hσµ, σν , σρi0,3,dσ = (−1) hσµ∆, σν , σρ∆i (1) ∨ (k) ∨ , 1 k! d1(σ1 ) +···+dk(σ1 ) d1+···+dk=d • 16 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

∨ (i) ∨ (i) where d, d1, . . . , dk ≥ 0 and σ1 (resp. (σ1 ) ) is the Poincaré dual homology class of σ1 (resp. σ1 ). This is easily proved using the Vafa-Intriligator residue formula (see [Ber96]). Equation (3.4) is an immediate consequence of (3.3). Notice that, for ` = 1, the equation (3.4) can be rewritten in the form k X j ◦ ϑ(σµ ∗G,q σ1) = σµ1+k−1 ∧ · · · ∧ σµi+k−i+1 ∧ · · · ∧ σµk i=1 i-th k−1 + (−1) δn−1,µ1+k−1σ0 ∧ σµ2+k−2 ∧ · · · ∧ σµk . The ﬁrst term coincides with the classical one, whereas the second term is the quantum correction dictated by the Quantum Pieri Formula (see [Ber97]). The following Proposition is an immediate consequence of the previous result. 2 2 Proposition 3.2. At the point p = t σ1 ∈ H (G; C) of the small quantum cohomology of G, the eigenvalues of the operator G • • Up := c1(G) ∗q (−): H (G; C) → H (G; C) are given by the sums

ui1 + ··· + uik , 1 ≤ i1 < ··· < ik ≤ n, where u1, . . . , uk are the eigenvalues of the corresponding operator U P for projective spaces at the point 2 2 pˆ := t σ1 + (k − 1)πiσ1 ∈ H (P; C), i.e. P • • Upˆ := c1(P) ∗(−1)k−1q (−): H (P; C) → H (P; C). • Proof. Let ϕ1, . . . , ϕn be the idempotents vectors of QH(−1)k−1q(P) and let

P Upˆ (ϕi) = ui · ϕi, i = 1, . . . , n. Then −1 j ◦ ϑ (ϕi1 ∧ · · · ∧ ϕik ), G with 1 ≤ i1 < i2 < ··· < ik ≤ n, are eigenvectors of Up with corresponding eigenvalues ui1 +···+uik .

4. Frequency of Coalescence Phenomenon in QH•(G(k, n)) Given 1 ≤ k < n, the canonical coordinates of the quantum cohomology of the projective space n−1 PC 2 2 n−1 at the point pˆ = t σ1 + (k − 1)πiσ1 in the small locus H ( ; ) are PC C 2 t + (k − 1)πi h−1 2πi u = n exp ζ , ζ := e n , h = 1, . . . , n. (4.1) h n n n Consequently, by Proposition 3.2, the canonical coordinates of the quantum cohomology of the Grass- 2 mannian G(k, n) at the point p = t σ1 are given by the sums k t2 + (k − 1)πi X n exp ζij , (4.2) n n j=1 for all possible combinations 0 ≤ i1 < i2 < ··· < ik ≤ n − 1. This means that, although general results guarantees the semisimplicity of the small quantum cohomology of Grassmannians (see Remark 2.3), it may happens that some Dubrovin canonical coordinates coalesce (i.e. the spectrum of the operator 2 c1(G(k, n)) ∗p (−) is not simple). More precisely, if there is a point p ∈ H (G(k, n); C) with coalescing canonical coordinates then all points of the small quantum locus have this property. In such a case, we will simply say that the Grassmannian G(k, n) is coalescing. In this and in the next sections, we want to answer to the following

Question 1. For which k and n the Grassmannian G(k, n) is coalescing? Question 2. How much frequent is this phenomenon of coalescence among all Grassmannians? For the answers we need some preliminary results. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 17

4.1. Results on vanishing sums of roots of unity. In this section we collect some useful notions and results concerning the problem of vanishing sums, and more general linear relations among roots of unity. The interested reader can ﬁnd more details and historical remarks in [Man65], [CJ76], [LJ78], [Zan89], [Zan95] and the references therein. Following [Man65] and the survey [LJ78], we will say that a relation k X aν zν = 0, aν ∈ Q, (4.3) ν=1 and zν ’s are roots of unity is irreducible if no proper sub-sum vanishes; this means that there is no relation k X bν zν = 0, with bν (aν − bν ) = 0 for all ν = 1, . . . , k ν=1 with at least one but all bν = 0. ∗ Theorem 4.1 (H.B. Mann, [Man65]). Let z1, . . . , zr be roots of unity, and a1, . . . , ar ∈ N such that r X aizi = 0. i=1 Moreover, suppose that such a vanishing relation is irreducible. Then, for any i, j ∈ {1, . . . , r} we have m zi Y = 1, m := p. z j p prime p≤r

Let G = hai be a cyclic group of order m, and let ζm be a ﬁxed primitive m-th root of unity. There is a well deﬁned natural morphism of ring

φ: ZG → Z[ζm]: a 7→ ζm, so that, we have the following identiﬁcation Z-linear relations among ker φ ≡ . the m-th roots of unity Let us also introduce

• the function ε0 : ZG → Z, deﬁned by   X ε0  xgg := card({g : xg 6= 0}); g∈G • a natural partial ordering on ZG, by declaring that given two sums X X x = xgg, y = ygg, g∈G g∈G

we have x ≥ y if and only if xg ≥ yg for all g ∈ G. We deﬁne NG := {x ∈ ZG: x ≥ 0}.

Theorem 4.2 (T.Y. Lam, K.H. Leung, [LL00]). Suppose that G is a cyclic group of order m = p1p2 . . . pr, with p1 < p2 < ··· < pr primes and r ≥ 2. Let φ: ZG → Z[ζm] be the natural map, and let x, y ∈ NG such that φ(x) = φ(y). If ε0(x) ≤ p1 − 1, then we have (A) either y ≥ x, (B) or ε0(y) ≥ (p1 − ε0(x))(p2 − 1).

In case (A), we have ε0(y) ≥ ε0(x), and in case (B) we have ε0(y) > ε0(x).

Corollary 4.1. In the same hypotheses of the previous Theorem, let us suppose that ε0(x) = ε0(y). Then x = y. • 18 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

Proof. We necessarily have case (A), and by symmetry of x and y, we conclude. Deﬁnition 4.1. Let n ≥ 2, and 0 ≤ k ≤ n. We will say that n is k-balancing if there exists a combination of integers 1 ≤ i1 < ··· < ik ≤ n such that

2πi i1 ik n ζn + ··· + ζn = 0, ζn := e . In other words, there are k distinct n-roots of unity whose sum is 0.

α1 αr Theorem 4.3 (G. Sivek, [Siv10]). If n = p1 . . . pr , with pi’s prime and αi > 0, then n is k-balancing if and only if {k, n − k} ⊆ Np1 + ··· + Npr. 4.2. Characterization of coalescing Grassmannians. Using the results exposed above on vanishing sums of roots of unity, we want to study and quantify the occurrence and the frequency of the coalescence of Dubrovin canonical coordinates in small quantum cohomologies of Grassmannians. Our ﬁrst aim is to explicitly describe the following sets, deﬁned for n ≥ 2:

An := {h: 0 < h < n s.t. G(h, n) is coalescing} , together with their complements

Aen := {h: 0 < h < n s.t. G(h, n) is not coalescing} . We need some previous Lemmata. Lemma 4.1. The following conditions are equivalent

• k ∈ An; • there exist two combinations

1 ≤ i1 < ··· < ik ≤ n and 1 ≤ j1 < ··· < jk ≤ n,

with ih 6= jh for at least one h ∈ {1, . . . , k}, such that

2πi i1 ik j1 jk n ζn + ··· + ζn = ζn + ··· + ζn , ζn := e .

Proof. It is an immediate consequence of Proposition 3.2, and formulae (4.1), (4.2). Lemma 4.2.

(1) If n is prime, then An = ∅. (2) If k ∈ {2, . . . , n − 2} is such that k ∈ An, then {min(k, n − k),..., max(k, n − k)} ⊆ An. (3) If n is k-balancing (with 2 ≤ k ≤ n − 2), then k ∈ An. Thus, if P1(n) ≤ n − 2, we have {P1(n), . . . , n − P1(n)} ⊆ An. Proof. Point (1) follows from Corollary 4.1. For the point (2), notice that given a linear relation as in Lemma 4.1 with k roots on both l.h.s. and r.h.s. we can obtain a relation with more terms, by adding to i1 ik i1 ik both sides the same roots. For point (3), if we have ζn + ··· + ζn = 0, then also ζn · (ζn + ··· + ζn ) = 0, and Lemma 4.1 applies. The last statement follows from the previous Theorem 4.3 and point (1).

Proposition 4.1. If P1(n) ≤ n − 2, then min An = P1(n).

Proof. Let k := min An. We subdivide the proof in several steps. • Step 1. Let us suppose that n is squarefree. By a straightforward application of Corollary 4.1, from an equality like

i1 ir j1 jr ζn + ··· + ζn = ζn + ··· + ζn ,

and r < P1(n) we deduce that necessarily ih = jh for all h = 1, . . . , r. Thus k = P1(n). This proves the Proposition if n is squarefree. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 19

• Step 2. From now on, n is not supposed to be squarefree. We suppose, by contradiction, that k < P1(n). Because of the minimality condition on k, in an equality

i1 ik j1 jk ζn + ··· + ζn = ζn + ··· + ζn , (4.4)

we have that ih 6= jh for all h = 1, . . . , k. Multiplying, if necessary, by the inverse of one root of unity, we can suppose that one root appearing in (4.4) is 1. Moreover, we can rewrite equation (4.4) as a vanishing sum 2k X αizi = 0, αi ∈ {−1, +1} (4.5) i=1 and where z1, . . . , z2k are distinct n-roots of unity. • Step 3. We show that the vanishing sum (4.5) is irreducible. Indeed, if we consider the smallest (i.e. with the least number of terms) proper vanishing sub-sum, then it must have at most k addends, otherwise its complement w.r.t. (4.5) would be a vanishing proper sub-sum with less terms. By application of Theorem 4.1 to this smallest sub-sum, we deduce that for all roots zi’s appearing in it, we must have m zi Y = 1, m := p. z j p prime p≤k

Under the assumption k < P1(n), we have that gcd(m, n) = 1, and since also z n z i = 1, we deduce i = 1, zj zj which is absurd by minimality of k. Thus (4.5) is irreducible. • Step 4. We now show that the order of any roots appearing in (4.5) must be a squarefree number. By application of Theorem 4.1, we know that for all i, j m zi Y = 1, m := p. z j p prime p≤2k m Since for one root in (4.5) we have zj = 1, we deduce that zi = 1 for any roots in (4.5), and that any orders, being divisors of m, must be squarefree. • Step 5. By applying the argument of Step 1, we conclude.

Theorem 4.4. The complex Grassmannian G(k, n) is coalescing if and only if P1(n) ≤ k ≤ n − P1(n). In particular, all Grassmannians of proper subspaces of Cp, with p prime, are not coalescing. Proof. The proof directly follows from Lemma 4.2 and Proposition 4.1. 4.3. Dirichlet series associated to non-coalescing Grassmannians, and their rareness. Let us now define the sequence lñ := card Aen , n ≥ 2. Introducing the Dirichlet series ∞ X lñ Le (s) := , ns n=2 we want deduce information about (˜ln)n≥2 studying properties of the generating function Le (s).

Theorem 4.5. The Dirichlet series Le (s) associated to the sequence (˜ln)n≥2 is absolutely convergent in the half-plane Re(s) > 2, where it can be represented by the inﬁnite series X p − 1 2ζ(s) Le (s) = − 1 . ps ζ(s, p − 1) p prime • 20 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

The function deﬁned by Le (s) can be analytically continued into (the universal cover of) the punctured half-plane ( ) ρ ρ pole or zero of ζ(s), {s ∈ : Re(s) > σ}\ s = + 1: , C k k squarefree positive integer

  1  X  3 σ := lim sup · log  l˜k , 1 ≤ σ ≤ , n→∞ log n 2  k≤n  k composite having logarithmic singularities at the punctures. In particular, at the point s = 2 the following asymptotic estimate holds 1 Le (s) = log + O(1), s → 2, Re(s) > 2. (4.6) s − 2

Proof. Let σa be the abscissa of (absolute) convergence for Le (s). Since

α inf {α ∈ R:l ˜n = O(n )} = 1, we have 1 ≤ σa ≤ 2. Moreover, the sequence (˜ln)n≥2 being positive, by a Theorem of Landau ([Cha68], [Ten15]) the point s = σa is a singularity for Le (s). For Re(s) > σa, we have (by Theorem 4.4)

X p − 1 X 2(P1(n) − 1) Le (s) = + . (4.7) ps ns p prime n composite Note that

X 2(P1(n) − 1) X X 2(p − 1) = ns (pm)s n composite p prime m≥2 P1(m)≥p   X p − 1  X 1  = 2 −1 +  ps ms p prime  m≥1  P1(m)≥p   ∞ X p − 1  Y X 1  = 2 −1 +  ps qks p prime  q prime k=0  q≥p   X p − 1 Y 1 = 2 −1 + ζ(s) 1 −  . ps  qs  p prime q prime q

From this and equation (4.7) it follows that

X p − 1 2ζ(s) Le (s) = − 1 . ps ζ(s, p − 1) p prime

ζ(s) Since for any s with Re(s) > 1 we have limn = 1, by asymptotic comparison we deduce that ζ(s,pn−1) the half-plane of absolute convergence of Le (s) coincides with the half-plane of ζP (s − 1) − ζP (s), hence σa = 2 ([Frö68]). • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 21

The second Dirichlet series in (4.7) deﬁnes an holomorphic function in the half-plane of absolute convergence Re(s) > σ, where ([HR15])   1  X  σ := lim sup · log  l˜k . n→∞ log n  k≤n  k composite

1 From the elementary and optimal inequality P1(n) ≤ n 2 , valid for any composite number n, we deduce 1 3 that 2 ≤ σ ≤ 2 . Thus, the sequence (αn)n∈N defined by   1  X  αn := · log  l˜k , log 2n  k≤2n  k composite is bounded: by Bolzano-Weierstrass Theorem, we can extract a subsequence converging to a positive real number r and, by characterization of the superior limit, we necessarily have r ≤ σ. Notice that we have the trivial estimate     X  X   X  l˜k =  l˜k +  l˜k > 2(n − 1), k≤2n 4≤k≤2n  k≤2n  k composite k even k odd composite 3 and we deduce that 1 ≤ r. In conclusion, 1 ≤ σ ≤ 2 . As a consequence, the function Le (s) can be extended by analytic continuation at least up to the half-plane Re(s) > σ, and it inherits from the function ζP (s − 1) − ζP (s) some logarithmic singularities in the strip σ < Re(s) ≤ 2: they correspond to the points of the form ρ + 1, 0 < Re(ρ) ≤ 1, k where ρ = 1 or ζ(ρ) = 0, and k is a squarefree positive integer. This follows from the well known representation ∞ X µ(n) ζ (s) = log ζ(ns), P n n=1 µ being the Möbius arithmetic function (see [Gle91], [Frö68] and [THB86]). For ρ = k = 1, we find again that s = 2 is a logarithmic singularity for Le (s): the asymptotic expansion (4.6) follows from 1 ζ (s) = log + O(1), s → 1, Re(ρ) > 1. P s − 1 This completes the proof. Corollary 4.2. The following statements are equivalent: 1 (1) (RH) all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(s) = 2 ; 0 (2) the derivative Le (s) extends by analytic continuation to a meromorphic function in the half-plane 3 2 < Re(s) with a single pole of oder one at s = 2. Remark 4.1. The analytic continuation of the function Le (s) beyond the line Re(s) = σ is highly influenced by the analytic continuation of the series

X l˜n ns n composite in the strip 1 < Re(s) < σ. In particular, if in this strip it does not have enough logarithmic singularities annihilating those of ζP (s − 1) − ζP (s), then the line Re(s) = 1 is necessarily a natural boundary for • 22 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

Le (s): indeed, the singularities of ζP (s − 1) cluster near all points of this line ([LW20]). Notice that s = σ is necessarily a singularity for Le (s), by Landau Theorem.

Remark 4.2. If we introduce the sequence ln := card (An), for n ≥ 2, and the corresponding generating function ∞ X ln L(s) := , ns n=2 the following identity holds: L(s) + Le (s) = ζ(s − 1) + ζ(s). In this sense, L(s) is “dual” to Le (s). Corollary 4.3. The following asymptotic expansion holds n X 1 n2 l˜ ∼ . k 2 log n k=2 In particular, the non-coalescing Grassmannians are rare: n 2 X lim l˜k = 0. n n2 − n k=2

Proof. Since the function Le (s) is holomorphic at all points of the line Re(s) = 2 but s 6= 2, and the asymptotic expansion (4.6) holds, an immediate application of Ikehara-Delange Tauberian Theorem for the case of singularities of mixed-type (involving both monomial and logarithmic terms in their principal parts) for Dirichlet series, gives the result (see [Del54] Theorem IV, and [Ten15] pag. 350). Another more elementary (and maybe less elegant) proof is the following: from Theorem 4.4 we have that n n X X l˜k = 2(1 − n) + π0(n) − π1(n) + 2 P1(j), k=2 j=2 and recalling the following asymptotic estimates (see [SZ68], [KL12] or [Jak13]) n1+α π (n) ∼ , α ≥ 0, α (1 + α) log n n X 1 nm+1 P (n)m ∼ , m ≥ 1, 1 m + 1 log n j=2 one concludes.

5. Distribution functions of non-coalescing Grassmannians, and equivalent form of the Riemann Hypothesis In this section we want to obtain some more ﬁne results about the distribution of these rare not coalescing Grassmannians. Thus, let us introduce the following

Deﬁnition 5.1. For all real numbers x, y ∈ R≥2, with x ≥ y, deﬁne the function

H(x, y) := card ({n ≤ x: n ≥ 2, l˜n > y}) . In other words, H is the cumulative number of vector spaces Cn, 2 ≤ n ≤ x, having more than y non-coalescing Grassmannians of proper subspaces. For x ∈ R≥4 we will deﬁne also the restriction 1 Hb (x) := H(x, 2x 2 ). In the following result, we describe some analytical properties of the function H. Theorem 5.1. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 23

(1) For any κ > 1, the following integral representation7 holds " ! # 1 Z ζ(s) xs H(x, y) = − 1 − ζP,y+1(s) + ζP, y +1(s) ds, 2πi y 2 s Λκ ζ s, 2 + 1

valid for x ∈ R≥2 \ N, y ∈ R≥2 (with y ≤ x), and where Λκ := {κ + it: t ∈ R} is the line oriented from t = −∞ to t = +∞. (2) For any κ > 1, the following integral representation holds " ! # 1 Z ζ(s) (y + 2)s ds H(x, y) = − 1 xs + ζ (s) − (y + 1)s , 2πi y P 2s s Λκ ζ s, 2 + 1

valid for x, y ∈ R≥2 \ N (with y ≤ x), and where Λκ := {κ + it: t ∈ R} is the line oriented from t = −∞ to t = +∞. (3) The following asymptotic estimate holds uniformly in the range x ≥ y ≥ 2 x γ log x 1 y H(x, y) = 1 e ω + O + O . ζ 1, 2 y + 1 log y log y log y Proof. The crucial observation is the following: if we consider, for ﬁxed x and y, the sets

A := {n: 2 ≤ n, l˜n > y} ,

B := {n: 2 ≤ n, 2P1(n) − 2 > y} , C := {p prime : p − 1 ≤ y, 2p − 2 > y} , then we have C ⊆ B and A ≡ B \ C. In this way:

• the Dirichlet series associated to the sequence 1A(n) (indicator function of A) is the difference of the Dirichlet series associated to 1B(n) and 1C(n). The first one is given by (see e.g. [Ten15]) ζ(s) y − 1, ζ s, 2 + 1 while the second one is given by the difference of partial sums

ζP,y+1(s) − ζ y (s). P, 2 +1 An application of Perron Formula for x not integer gives the integral representation (1) of P 1 n≤x A(n). • Moreover, we also get the identity y y H(x, y) = Φ x, + 1 − π (y + 1) − π + 1 . (5.1) 2 0 0 2 For x and y not integer, we can apply Perron Formula separately for the three terms: ! y 1 Z ζ(s) xs Φ x, + 1 = − 1 ds, 2 2πi y s Λκ ζ s, 2 + 1 Z 1 s ds π0(y + 1) = ζP (s)(y + 1) , 2πi Λκ s y 1 Z y s ds π0 + 1 = ζP (s) + 1 . 2 2πi Λκ 2 s The sum of the three terms gives the second integral representation (2). • Form equation (5.1), by applying the well known de Bruijn’s asymptotic estimate ([dB50], [SMC06]), we obtain the estimate (3).

7The integral must be interpreted as a Cauchy Principal Value. • 24 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS

Theorem 5.2. The function Hb admits the following asymptotic estimate: Z x dt Θ Hb (x) = + O x log x , where Θ := sup {Re(ρ): ζ(ρ) = 0} . 0 log t Hence the following statements are equivalent: 1 (1) (RH) all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(s) = 2 ; (2) for a suﬃciently large x, the following (essentially optimal) estimate holds x Z dt 1 Hb (x) = + O x 2 log x . (5.2) 0 log t 1 Proof. Using the elementary fact that for any composite number n we have P1(n) ≤ n 2 , we obtain the estimate 1 1 Φ(x, x 2 ) = π0(x) + O x 2 .

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